Five false threshold tables in Cohen's conjectures on primes and cyclic numbers

Abstract

A positive integer n is cyclic if every group of order n is cyclic, equivalently (Szele) if gcd(n,φ(n))=1. Cohen's Conjectures about primes and cyclic numbers collects sixty-three conjectures: cyclic-number analogues of classical conjectures about primes, together with new conjectures about the primes themselves. Many of them have a two-part shape: an asymptotic existence clause "for every k there is a threshold N(k) such that …", followed by a printed table of specific values of N(k). The existence clauses are out of reach; the tables are finite claims. We show that five of those tables are false: Conjectures 17, 20, 22, 28 and 29 fail at thirty-nine of their printed constants, and we exhibit an explicit witness for each. Four of the five refutations follow from counts that the source prints itself, in one case three lines below the conjecture they contradict, and in another from a sequence in the OEIS that predates the conjecture by nine years; we recompute all of them from the definitions rather than quoting them. The refutation of Conjecture 22 is horizon-free: because its threshold is defined to be least, one count at one index suffices, with no claim about large arguments. We also show that three further printed thresholds can be lowered unconditionally. What is not refuted, in every case, is the existence clause; we separate the two halves explicitly, because the recent literature reports Conjectures 17 and 20 as proved, and the only statement that literature prints for either conjecture is the existence clause alone. All statements are verified in Lean 4 by the kernel, with no compiled evaluation anywhere.

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